The nearby, middle-aged PSR B1055-52 has many properties in common with the Geminga pulsar. Motivated by the Geminga's enigmatic and prominent pulsar wind nebula (PWN), we searched for extended emission around PSR B1055-52 with Chandra ACIS. For an energy range 0.3-1 keV, we found a 4σ flux enhancement in a 4".9-20" annulus around the pulsar. There is a slight asymmetry in the emission close, 1".5-4", to the pulsar. The excess emission has a luminosity of about 1029 erg/s in an energy range 0.3-8 keV for a distance of 350 pc. Overall, the faint extended emission around PSR B1055-52 is consistent with a PWN of an aligned rotator moving away from us along the line of sight with supersonic velocity, but a contribution from a dust scattering halo cannot be excluded. Comparing the properties of other nearby, middle-aged pulsars, we suggest that the geometry–the orientations of rotation axis, magnetic field axis, and the sight-line–is the deciding factor for a pulsar to show a prominent PWN. We also report on an ≳ 30% flux decrease of PSR B1055-52 between the 2000 XMM-Newton and our 2012 Chandra observation. We tentatively attribute this flux decrease to a cross-calibration problem, but further investigations of the pulsar are required to exclude actual intrinsic flux changes.